Prime numbers dividing infinitely many numbers in an sequence

Here I found a question:

Show that every prime not equal to $2$ or $5$ divides infinitely many
of the numbers $1,11,111,1111,dots$ etc.

which is partly solved here Prime numbers divide an element from a set.

From this the following conjecture seems reasonable:

Given any finite set $S={q_1,dots,q_k}$ of $k$ primes, then any
prime $pnotin S$ divides infinitely many of the numbers
$a_1,a_2,a_3dots$, where $a_1=1$ and $a_{n+1}=1+a_nprod q_i$.

Can this be proved?