The des-Wilf equivalences among four-letter non-classical patterns

For a permutation pattern $\tau$, let $F_n^{\operatorname{des}}(\tau;q)$ denote the descent generating polynomial over permutations in $\operatorname{Av}_n(\tau)$, and write $\tau\equiv_{\operatorname{des}}\rho$ when $F_n^{\operatorname{des}}(\tau;q)=F_n^{\operatorname{des}}(\rho;q)$ for every $n$. The des-Wilf equivalences conjecture. The following equivalences hold:

$$1243\equiv_{\operatorname{des}}3412\qquad\text{and}\qquad 1342\equiv_{\operatorname{des}}2413.$$

These conjectured equalities concern descent-preserving enumeration of permutations avoiding the displayed non-classical patterns; the paper reports supporting data but no closed forms for the relevant generating polynomials are known.