The refined low-degree conjecture for polynomial-time distinguishability

Let $(\mathsf{P}_n)$ and $(\mathsf{Q}_n)$ be sufficiently nice sequences of probability measures, and let $L_n^{\leq D}$ be the degree-$D$ low-degree likelihood ratio. Strong distinguishability means that a polynomial-time algorithm can reliably distinguish samples from the two sequences.

Refined low-degree conjecture. If there exists a polynomial-time algorithm that strongly distinguishes $(\mathsf{P}_n)$ and $(\mathsf{Q}_n)$, then

$$\|L_n^{\leq D}\|=\omega(1)$$

for every $D=\omega(1)$.

This is a stronger empirical refinement of the informal low-degree conjecture: it removes the extra logarithmic factor in the degree threshold. The paper reports supporting evidence across several high-dimensional inference problems, but does not establish the assertion in general.