Jet closure formula for weighted homogeneous polynomials

Let $k$ be an algebraically closed field of characteristic $0$, let $R=k[[x_{1},...,x_{n}]]$ with maximal ideal $\mathbf{m}$, and let $f\in R$ be a weighted homogeneous polynomial. Write $J(f)$ for the Jacobian ideal of $f$, and let $J(f)^{m-jc}$ denote its $m$-th jet closure. Jet closure conjecture.

$$J(f)^{m-jc}=J(f)+\mathbf{m}^{m+1}.$$

This conjecture extends the explicit homogeneous case proved earlier in the paper. The authors report that the formula holds in computations for weighted homogeneous polynomials in $k[[x,y]]$, but no proof or resolution is supplied here.