Exponential waiting times question for two processes and showing with probability $1$ that after some time only one process occurs?

Two companies compete for customers. Customers arrive one at a time, and each buys either Company $1$’s product (denoted by “iPhone”) or Company $2$’s product (denoted by “Android phone”), but not both. Initially, each company has had $1$ sale. Each customer chooses which product to buy with probability proportional to the square of the number of sales, e.g., (if $4$ iPhones and $3$ Android phones have been sold, then the next customer buys an iPhone with probability $\frac{16}{25}$ and an Android phone with probability $\frac{9}{25}$).

I'd like to show with probability $1$, either there is a customer after whom all future customers buy iPhones, or there is a customer after whom all future customers buy Android phones.

My approach is to let $X_1,X_2,\ldots$ denote the interarrival times between purchases of iPhones. We assume that they have distribution modeled by $X_j \sim \frac{1}{j^2}Expo$. In a similar way we define the Android buying interarrival times to be $Y_1,Y_2,\ldots$ which are distributed $Y_i \sim \frac{1}{i^2}Expo$. At this point, I'm not sure how to go further, does anyone have any ideas?

The interarrival times are completely irrelevant to your question. All that matters is the history of outcomes; i.e., you should define a process $X_{n+1} \mid X_1, X_2, \ldots, X_n \sim \operatorname{Bernoulli}(p_{n+1})$, where $p_{n+1}$ is a suitably chosen function of some sufficient statistic (hint, hint) of the previous outcomes $X_1, X_2, \ldots, X_n$.