The scaled generalization of Tepper's identity

Let $l$ be a positive integer, and let $P(x)$ be a real polynomial of degree $n$ with leading coefficient $a_n$. Scaled generalized Tepper identity. One has

$$\sum_{k=0}^{n}(-1)^k {n \choose k}P(x-lk)=a_n l^n n!.$$

The preceding theorem proves the case $l=1$ for arbitrary real polynomials. The displayed statement is presented as a conjectural generalization; the supplied text gives no resolution evidence for the scaled case.