Browning–Van Valckenborgh prediction for squareful triples

Let $N_1(B)$ denote the counting function for primitive triples of nonzero squareful integers in the setting above, and let $c_{\textrm{BV}}$ be the constant obtained by summing the constants from Manin's conjecture for the conics

$$x_0^2y_0^3+x_1^2y_1^3=x_2^2y_2^3.$$

Browning–Van Valckenborgh prediction. One has

$$N_1(B)\sim 3c_{\textrm{BV}}B^{1/2}.$$

This is the $n=1$ prediction for the asymptotic count of squareful solutions. The constant is given explicitly in the cited work and arises from the corresponding sum of Manin-type constants; the source does not state whether the prediction has been resolved.