Sign and vanishing conjecture for boundary strata of Reeb flows

Let $\beta$ be a smooth contact form on a compact connected manifold $X$, and let $v_\beta$ be its boundary-generic Reeb vector field. For the boundary strata $\partial_j^\pm X(v_\beta)$, the following assertions are conjectured.

Boundary-strata conjecture. If $j$ is even, then $\beta|_{\mathsf{\int}(\partial_j^+X(v_\beta))}$ is a contact form away from a set of measure zero, and

$$\mp\left[\beta\wedge(d\beta)^{\frac{2n-j}{2}}\right]\big|_{\partial_j^\pm X(v_\beta)}\geq 0,$$

while

$$\left.\left.\beta\wedge(d\beta)^{\frac{2n-j}{2}}\right|\right|_{\partial_{j+1}X(v_\beta)}\equiv 0.$$

If $j$ is odd, then $d\beta|_{\mathsf{\int}(\partial_j^+X(v_\beta))}$ is symplectic away from a set of measure zero, and

$$\pm(d\beta)^{\frac{2n+1-j}{2}}\big|_{\partial_j^\pm X(v_\beta)}\geq 0,$$

while

$$\left.\left.(d\beta)^{\frac{2n+1-j}{2}}\right|\right|_{\partial_{j+1}X(v_\beta)}\equiv 0.$$

These assertions predict a systematic contact or symplectic structure, sign condition, and vanishing condition on the successive boundary strata of a boundary-generic Reeb flow. The source gives examples motivating the claim but no resolution evidence.