Minimal polynomial conjecture for orientation-reversing stretch factors on odd-genus orientable surfaces

Let $S_g$ be the closed orientable surface of genus $g$, and let $\delta^+_{\mathrm{rev}}(S_g)$ be the minimal stretch factor among orientation-reversing pseudo-Anosov homeomorphisms of $S_g$ with orientable invariant foliations. For a real number $r>1$, call it the largest root of a polynomial if it is the greatest real root of that polynomial.

Odd-genus orientation-reversing stretch factor conjecture. For all $k\geq 2$, $\delta^+_{\mathrm{rev}}(S_{2k-1})$ is the largest root of

$$x^{4k}-x^{2k+1}-x^{2k-1}-1.$$

The conjecture follows the computed values for odd genera through $11$ and gives a uniform formula for the odd-genus sequence. The general statement remains open.