Nonsmooth splitting conjecture for infinitesimally Minkowskian TCD spaces

Let $(\mathsf{M},l,\mathfrak{m})$ be a metric measure spacetime. It is infinitesimally Minkowskian if, for all $l$-causal functions $u$ and $v$, its maximal weak subslope satisfies

$$2\lvert\mathrm{d}u\rvert_*^2+2\lvert\mathrm{d}(u+v)\rvert_*^2=\lvert\mathrm{d}v\rvert_*^2+\lvert\mathrm{d}(2u+v)\rvert_*^2\quad\mathfrak{m}\text{-a.e.}$$

In particular, an $\mathsf{LTCD}_q^e(K,N)$ space is an infinitesimally Minkowskian $\mathsf{TCD}_q^e(K,N)$ space. Nonsmooth splitting conjecture. Let $(\mathsf{M},l,\mathfrak{m})$ be a globally hyperbolic $\mathsf{LTCD}_q^e(0,N)$ metric measure spacetime, where $N$ is a real number no less than one. If it contains a line, then it is isomorphic as a metric measure spacetime to the generalized cone

$$(\mathbb{R}\times \mathsf{N},\lvert\cdot-\cdot\rvert\times \mathsf{d},\mathcal{L}^1\otimes\mathfrak{n}),$$

where $(\mathsf{N},\mathsf{d},\mathfrak{n})$ is an $\mathsf{RCD}(0,N-1)$ metric measure space. This is intended as a nonsmooth Lorentzian analogue of splitting theorems for RCD spaces. General TCD spaces do not split because Finsler spacetimes provide counterexamples, while the additional infinitesimal Minkowskianity condition is designed to recover a Lorentzian splitting result. The source does not indicate that this statement has been proved.