Equilibrium-count conjecture for symmetric finite trails

Consider the Trail of Lost Pennies on the finite trail $\llbracket -j-1,k+1\rrbracket$ in its symmetric standard form. Finite-trail equilibrium-count conjecture. The number of time-invariant Nash equilibria equals

$$\max\{2(j+k)-5,1\}.$$

The formula is presented as a conjecture about the effect of reflection symmetries on the finite-trail game; no proof or resolution is supplied in the text.