The Brownian-sheet small deviation conjecture

Let $B$ be the Brownian sheet on $[0,1]^d$, viewed as a centered Gaussian process with covariance

$$\mathbb E[B_sB_t]=\prod_{j=1}^d\min(s_j,t_j).$$

Brownian-sheet small deviation conjecture. For $d\ge2$, as $\epsilon\downarrow0$,

$$-\log\mathbb P\left(\|B\|_{C([0,1]^d)}<\epsilon\right)\simeq\epsilon^{-2}(\log(1/\epsilon))^{2d-1}.$$

By the Kuelbs--Li correspondence, this is the probabilistic form of the metric entropy conjecture. The claimed exponent is conjectural in the general dimensions considered here and follows from the corresponding entropy asymptotics if those are established.