For $q$ a prime power and $t \geq 0$ let $a_t^q=\sum\limits_{k=0}^{t}{[t,k]_q}$ with $[t,k]_q$ the Gaussian binomial coefficient, see https://en.wikipedia.org/wiki/Gaussian_binomial_coefficient.

Question: Does there exist $t \geq 1$ and $q$, such that $a_t^q+1$ has at least $t+1$ prime divisors counted with multiplicity?

In case I made no mistake (but it is late here...), a positive answer to this question would give a positive answer to the question of Jeremy Rickard in The number of ideals in a ring .