Numerical evidence indicates that Jacobi polynomials with negative integer parameters satisfy the identity
$$P_n^{(-m,-k)}(x)=\left(\frac{x-1}{2}\right)^m\left(\frac{1+x}{2}\right)^kP_{n-m-k}^{(m,k)}(x),$$ where $n\ge m+k$. How this identity can be proved?

I came across this supposed identity when comparing some linearisation  relations for the product of two Laguerre polynomials from https://cds.cern.ch/record/1378854?ln=en (after correcting typos) and https://iopscience.iop.org/article/10.1088/0305-4470/18/9/022 but such a proof (assuming both papers are correct) is too complicated and I prefer a direct proof which then I can use to argue that the results of these two papers are equivalent to each other.