Prime smoothness and prescribed odd-prime valuations

Let $n>0$ be an integer and let $x\in\{0,1\}^n$ be a vector of bits of length $n$. For each $1\leq k\leq n$, let $q_k$ denote the $k$th odd prime. Prime smoothness and prescribed valuations conjecture. There is a prime $p$ such that $p-1$ is $\operatorname{poly}(n)$-smooth and, for every $1\leq k\leq n$, the $k$th odd prime $q_k$ divides $p-1$ exactly $x_k$ times.

This conjecture supplies primes whose factorization has both controlled smoothness and prescribed multiplicities for the first $n$ odd primes, as required by the paper's number-theoretic construction. Its status is not resolved in the supplied text.