Rigidity conjecture for spacelike entire self-shrinking graphs

Let $u=(u^1,u^2,\ldots,u^n)$ be an entire smooth solution of

\sum_{i,j=1}^m g^{ij}(x)u^_{ij}(x)=\frac12\left(\sum_{i=1}^m x_i u^_i(x)-u^(x)\right),\qquad x\in\mathbb{R}^m,\quad \alpha=1,\ldots,n,

where

$$g_{ij}(x)=\delta_{ij}-\sum_{\alpha=1}^n u^\alpha_i(x)u^\alpha_j(x),$$

and $(g^{ij}(x))_{1\leq i,j\leq m}$ is the inverse matrix of $(g_{ij}(x))_{1\leq i,j\leq m}$. Assume $u^1(0)=\cdots=u^m(0)=0$ and

$$\liminf_{|x|\to\infty}\frac{\log\det(g_{ij}(x))}{|x|^2-|u(x)|^2}\geq-\frac12.$$

Rigidity conjecture. Then $u^\alpha(x)$ is a linear function for each $\alpha=1,\ldots,n$.

This proposes a rigidity statement for entire spacelike self-shrinking graphs in pseudo-Euclidean space, motivated by the one-dimensional example and the preceding corollary. The supplied text does not indicate whether the conjecture has been resolved.