Equidistribution of values of linear forms on a cubic hypersurface

Let $C$ be a cubic form with rational coefficients in $n$ variables, and let $h$ be the $h$-invariant of $C$. Let $L_1, \ldots, L_r$ be linear forms with real coefficients such that if $\boldsymbol{\alpha} \in \mathbb{R}^r \setminus \{ \boldsymbol{0} \}$ then $\boldsymbol{\alpha} \cdot \mathbf{L}$ is not a rational form. Assume that $h > 16 + 8 r$. Let $\boldsymbol{\tau} \in \mathbb{R}^r$, and let $\eta$ be a positive real number. We prove an asymptotic formula for the weighted number of integer solutions $\mathbf{x} \in [-P,P]^n$ to the system $C(\mathbf{x}) = 0, \: |\mathbf{L}(\mathbf{x}) - \boldsymbol{\tau}| < \eta$. If the coefficients of the linear forms are algebraically independent over the rationals, then we may replace the $h$-invariant condition with the hypothesis $n > 16 + 9 r$, and show that the system has an integer solution. Finally, we show that the values of $\mathbf{L}$ at integer zeros of $C$ are equidistributed modulo one in $\mathbb{R}^r$, requiring only that $h > 16$.