An Exponential Diophantine Equation related to the Erdos Ternary Conjecture

I am studying my conjecture that $$2^n$$ and $$2^{n+1}$$ have a common digit in base 5 if $$n>6$$. I believe that this conjecture is true provided that

$$2^x=12(5^{y_1}+ \dots +5^{y_k}) + 5^z - 1$$

has no solutions in integers where $$x \ge 0$$, $$0\le y_1 < \dots , and $$z \ge 0$$. This arises from the fact that, in base 5, $$2 \times 1 = 2$$ and $$2 \times 14_5 = 33_5$$; so that for example $$2 \times 114111141411_5 = 233222333322_5$$. The first difficult base is 5: bases 2 and 4 are trivial and base 3 can be solved.

This is similar to the unsolved Erdos ternary conjecture that involves

$$2^x=3^{y_1}+ \dots +3^{y_k}.$$

My question is whether these two exponential Diophantine equations are related so is equally difficult to solve, or whether other techniques (such as modular considerations) can help to solve it.