A question Bounds for constructing $n!$ with additions, subtractions, and multiplications starting from $1$ was asked on constructing $ak!$ with ring operations.
$\tau$ conjecture states if $\exists$ no infinite sequence of integers $n_k$ such that $k!n_k$ could be constructed in polylogarithmic number of ring operations, then $P_C\neq NP_C$ in BSS model.
Is there any consequence if $\tau$ conjecture fails?
