Second-order Eulerian formula conjecture for flattened Fubini rankings

Let $\mathrm{FR}_n$ denote the set of Fubini rankings on $n$ competitors, and let $\mathrm{flat\_runs}_k(\mathrm{FR}_n)$ denote the flattened Fubini rankings with $k$ runs of ascents. Define

$$s(n,k)=|\mathrm{flat\_runs}_k(\mathrm{FR}_n)|.$$

Let $E_2(a,b)$ denote the second-order Eulerian numbers. Second-order Eulerian formula conjecture. For every relevant integer $j$,

$$s(2j+1,j-1)=\sum_{i=0}^{j-1}E_2(2j+1,i)2^i.$$

The source identifies the second-order Eulerian numbers with OEIS A340556 and states that proving this formula, as well as finding formulas for $s(n,k)$ in general, remains open.