Conjecture on explosive distances in scale-free percolation

Let $d_L(x,y)$ denote the $L$-distance between vertices $x,y\in\mathbb{Z}^d$, and let $V_x$ be the explosion time from $x$. Under the condition of the distance theorem, these quantities satisfy the following almost-sure bound. Explosive-distance conjecture. For any two vertices $x,y\in\mathbb{Z}^d$,

$$d_L(x,y)-V_x+V_y\leq 0$$

almost surely. Consequently, as $\|x\|\to\infty$,

$$\mathrm d_L(0,x)-(V_0+V_x)\buildrel {a.s.}\over{\longrightarrow}0.$$

The conjecture strengthens the asserted result on explosive $L$-distances, predicting that distances asymptotically decompose into the explosion times from the two endpoints. The supplied text gives no resolution, so its status remains open.