Is there a near-repdigit prime of every size? Define a near-repdigit prime as a prime number where all but one of the digits are the same. For example $199, 94999$  and $76777$ are near-repdigit primes.
Is there a near-repdigit prime with $n$ digits for all $n>1$?
I realise that a lot of these types of questions are open. Is this in fact an open question?

A stronger conjecture is the following. Does there in fact exist a near-repdigit prime with $n$ digits whose first digit is $1$ for all $n>1$?
I have checked for near-repdigit primes starting with the digit 1 and there is at least one for all $n \leq 3495$.

Following a suggestion of @ErickWong I counted how many counter-examples there are for both the stronger conjecture and the original conjecture  in base 2,3, 4 and 5 for $2 \leq n \leq 2000$.

*

*base 2: Stronger 269, original 269

*base 3: Stronger 58, original 58

*base 4: Stronger 3, original 3

*base 5: There are 0 counter-examples for both the stronger and the original conjecture.

For every case where there is no near-repdigit prime with the first digit being 1, there is no near-repdigit prime with that number of digits and in that base at all.
 A: The $n$-digit numbers of this form can be parametrized as $A\cdot\frac{10^n-1}{9} + (B-A)\cdot10^k$, where $k \in \{0,1,2,\ldots, n-1\}$, $A \in \{1,2,\ldots, 9\}$, and $B\neq A\in\{0,1,2,\ldots,9\}$.  So, as pointed out in comments, there are exactly $81n$ numbers of this form for a given $n$.  If there's nothing "special" about these numbers (nothing making them more or less likely to be prime except their size), then each is prime with probability about $1/(n\log 10)$, and the probability that none are prime, for large $n$, behaves like
$$
\left(1 - \frac{1}{n\log 10}\right)^{81n}\approx e^{-81/\log 10}\approx e^{-35.2} \approx 5.3\times10^{-16}.
$$
In other words, it's very likely for any particular $n$ that there's a near-repdigit prime of that length.  But (still assuming there's nothing special about the numbers), there should be infinitely many $n$ for which there are no near-repdigit primes of length $n$, and we can expect the smallest such $n$ to be of order $10^{15}$ or so.
Based on the OEIS sequence A258915, the actual number of near-repdigit primes is somewhat higher than expected for smallish $n$: we expect $35.2$ on average by the above analysis, but the actual mean is more like $44$.  This can most likely be explained by examining divisibility of these numbers by small primes.  For instance, while half of all numbers are divisible by $2$, only about $4/9$ of the numbers we're testing are (mainly those where $A$ is even), leading to a higher fraction of primes.  Similarly, one-fifth of all numbers are divisible by $5$, but only about $1/9$ of the numbers we're testing are (mainly those with $A=5$).  Taking this extra data into account, we might adjust our expectation quantitatively... the smallest counterexample may be more like $10^{19}$... but the qualitative conclusion is unchanged.

The stronger conjecture, that for each $n$ there is a near-repdigit prime of length $n$ whose first digit is a $1$, should be much easier to falsify.  The $n$-digit numbers of this form are those where either (a) $A=1$ and $k<n-1$ (all $1$s except a single digit), or (b) $B=1$ and $k=n-1$ (only the first digit is a $1$).  There are $9n$ of these numbers, making the naive probability that none are prime
$$
\left(1 - \frac{1}{n\log 10}\right)^{9n} \approx e^{-9/\log 10}\approx e^{-3.91} \approx 0.02.
$$
So you shouldn't have to check that many values of $n$ to find a counterexample.  The same reasoning applies for the other possible first digits; there should be abundant counterexamples for all of them.  As with the main problem, they are expected to be somewhat less abundant than this analysis suggests, but should still be in reach of a feasible computer search.

Update:
I've searched through $n=500$ for counterexamples to the stronger conjecture, both for numbers starting with $1$ (OP's conjecture) and for numbers starting with $3$, $7$, or $9$ (the other initial digits are easy to find counterexamples for, for obvious reasons) and haven't found any yet.  Empirically the number of near-repdigit primes (probable primes, in practice) for each of these starting digits averages $11$ or so, as compared to the expected $3.91$.  This makes the expected smallest counterexample more like $n=10^4$ or $10^5$.
