Conjectural product formula for normalized volumes of klt singularities

Let $(x_i i (X_i,D_i))$ be klt singularities for $i=1,2$, and write $n_i:=\dim X_i$. Let $\pi_i:X_1\times X_2\to X_i$ be the projections. Conjectural product formula.

$$\frac{\widehat{\mathrm{vol}}((x_1,x_2),X_1\times X_2,\pi_1^*D_1+\pi_2^*D_2)}{(n_1+n_2)^{n_1+n_2}}=\frac{\widehat{\mathrm{vol}}(x_1,X_1,D_1)}{n_1^{n_1}}\cdot\frac{\widehat{\mathrm{vol}}(x_2,X_2,D_2)}{n_2^{n_2}}.$$

The formula would express multiplicativity of normalized local volumes, after the dimension-dependent normalization, under products of klt singularities. It is suggested by the equality established for the product valuations in the preceding lemma; the general conjectural product formula remains open in the source.