Zheng's zero Laplacian eigenvalue multiplicity conjecture for uniform hypertrees

Let $T=(V(T),E(T))$ be a $k$-uniform hypertree for $k\geq 3$. The multiplicity of the zero Laplacian eigenvalue of $T$ is

$$k^{|E(T)|(k-2)}.$$

Zheng's conjecture. For every $k$-uniform hypertree $T$ with $k\geq 3$, the multiplicity of the zero Laplacian eigenvalue is $k^{|E(T)|(k-2)}$. This conjecture concerns the Laplacian spectrum of uniform hypertrees and extends the known computations for uniform hyperstars and hyperpaths; its resolution is not specified in the supplied text.