The conditional-coordinate-tail conjecture for isotropic down-monotone logconcave distributions

Let $f$ be an isotropic down-monotone logconcave function in $\mathbb{R}^m$, and let $X$ have density $f$. Let $1>p>0$. The conditional-coordinate-tail conjecture. There exists an absolute constant $c$ such that, for any $t\le m$,

$${\sf P}(X_t\ge p\mid X_1,\ldots,X_{t-1}\ge p)\le (1+cp^2){\sf P}(X_t\ge p).$$

This asserts that conditioning the preceding coordinates to exceed the same threshold increases the next-coordinate tail probability by at most a multiplicative factor $1+cp^2$; the supplied text gives no evidence that the claim has been resolved.