Free-boundary pseudocritical scaling conjecture for the weakly coupled $|\varphi|^4$ model

Consider the $|\varphi|^4$ model with free boundary conditions (FBC), and let $\chi_{g,\nu,N}^{\rm F}$ denote its susceptibility, while $\chi_{g,\nu,N}$ denotes the susceptibility in the periodic setting. Let $\nu_c$ be the critical parameter, and let $v_N$ and $w_N$ be the volume-dependent pseudocritical and critical-window scales, respectively. Free-boundary pseudocritical scaling conjecture. There exist $C,c,c'>0$ such that

$$\chi_{g,\nu_c-cv_N+sw_N,N}^{\rm F}\sim C\chi_{g,\nu_c+c'sw_N,N}.$$

This conjecture predicts that the standard finite-size scaling profile is recovered under free boundary conditions after a volume-dependent shift of the critical parameter. The analogous phenomenon is known in the hierarchical model, while the Euclidean case remains open and may involve different constants.