The local derivation conjecture for maximal solvable Lie algebras

Let $L$ be a maximal solvable Lie algebra with nilradical $N$. A local derivation on $L$ is a linear map such that, for every $x\in L$, there exists a derivation $D_x$ of $L$ with the same value at $x$. Local derivation conjecture. Any local derivation on $L$ is a derivation. The preceding examples establish this property for specific maximal solvable Lie algebras, but the general assertion is presented as a conjecture.