Existence of separating quadratic-residue tests for Mersenne factors

Let $p>p_0$ be prime, assume the distinct-factor bound conjecture and the uniform-distribution conjecture, and write

$$2^p-1=q_1\cdots q_k,$$

where $q_1<\cdots<q_k$ are the prime factors and $k<1.36\log p$. Let $\delta_{ij}$ denote the Kronecker delta. The separating-exponents conjecture. For every $i=1,\ldots,k$, there exists an odd integer $a_i\in[5,p-2]$ such that

$$\left(\frac{2^{a_i}+1}{q_j}\right)=(-1)^{\delta_{ij}}\qquad(1\le j\le k).$$

This would provide, for each prime factor $q_i$, an exponent whose value is a nonresidue modulo $q_i$ and a residue modulo every other prime factor; the source derives it heuristically from the preceding conjectures and an independence estimate.