Demaillys conjecture on Waldschmidt constants for sufficiently many very general points in $mathbb{P}^n$

Let $Z$ be a finite set of $s$ points in the projective space $mathbb{P}^n$ over an algebraically closed field $F$. For each positive integer $m$, let $alpha(mZ)$ denote the smallest degree of nonzero homogeneous polynomials in $F[x_0,ldots,x_n]$ that vanish to order at least $m$ at every point of $Z$. The Waldschmidt constant $widehat{alpha}(Z)$ of $Z$ is defined by the limit [ widehat{alpha}(Z)=lim_{m to infty}frac{alpha(mZ)}{m}. ] Demailly conjectured that [ widehat{alpha}(Z)geqfrac{alpha(mZ)+n-1}{m+n-1}. ] Recently, Malara, Szemberg, and Szpond established Demaillys conjecture when $Z$ is very general and [ lfloorsqrt[n]{s}rfloor-2geq m-1. ] Here we improve their result and show that Demaillys conjecture holds if $Z$ is very general and [ lfloorsqrt[n]{s}rfloor-2ge frac{2varepsilon}{n-1}(m-1), ] where $0le varepsilon<1$ is the fractional part of $sqrt[n]{s}$. In particular, for $s$ very general points where $sqrt[n]{s}inmathbb{N}$ (namely $varepsilon=0$), Demaillys conjecture holds for all $minmathbb{N}$. We also show that Demaillys conjecture holds if $Z$ is very general and [ sgemax{n+7,2^n}, ] assuming the Nagata-Iarrobino conjecture $widehat{alpha}(Z)gesqrt[n]{s}$.