An orbit of symmetric polynomials

Suppose $f(x,y,z)$ is symmetric (in the following, symmetric tout court always means "symmetric w.r.to the three variables $(x,y,z)$") . Then $\mathcal{L}f(x,y,z):=(x+y)(x+z)f(x-1,y,z)-x^2f(x,y,z)$ is already symmetric w.r.to $(y,z)$, so it is symmetric if and only if it is symmetric w.r.to $(x,y)$, that is, after simplifications, if and only if $f$ satisfies the functional equation $$(x-y)f(x,y,z)-(x+z)f(x-1,y,z)+(y+z)f(x,y-1,z)=0.$$

Claim: if both $f$ and $\mathcal{L}f$ are symmetric, so is $\mathcal{L}^2f$. In other words, the space of all symmetric solutions to the above functional equation is $\mathcal{L}$-invariant.

Proof: According to the observation above, in order to prove the claim, we need to check that the following is identically zero: $$(x-y)\mathcal{L}f(x,y,z)-(x+z)\mathcal{L}f(x-1,y,z)+(y+z)\mathcal{L}f(x,y-1,z) $$ and we exploit the symmetry of $\mathcal{L}f$ writing it $$(x-y)\mathcal{L}f(x,y,z)-(x+z)\mathcal{L}f(y,x-1,z)+(y+z)\mathcal{L}f(x,y-1,z) $$ namely $$(x-y)\big[(x+y)(x+z)f(x-1,y,z)-x^2f(x,y,z)\big]$$$$ -(x+z)\big[(y-1+x)(y+z)f(y-1,x-1,z)-y^2f(y,x-1,z)\big]$$$$ +(y+z)\big[(x+y-1)(x+z)f(x-1,y-1,z)-x^2f(x,y-1,z)\big]$$

$$=\big[(x-y) (x+y)(x+z) +(x+z) y^2 \big]f(x-1,y,z)$$$$-(x-y)x^2f(x,y,z) -(y+z) x^2f(x,y-1,z) =$$

$$=-x^2\big[(x-y)f(x,y,z)-(x+z)f(x-1,y,z)+(y+z)f(x,y-1,z)\big]$$

which is indeed zero, according to the initial remark, since $\mathcal{L}f$ was assumed symmetric.

Noting Pietro's finite check, I can report that all symmetric polynomials $p(x_1,x_2,x_3)$ up to degree 18 inclusive such that $\mathcal{L}p$ is symmetric are linear combinations of $1,\mathcal{L}1,\mathcal{L}^21,\ldots\,$.

This strongly suggests that there are no others.

An observation that might lead to an elementary proof is that, up to degree 18, all polynomials $p$ such that both $p$ and $\mathcal{L}p$ are symmetric are uniquely determined by the coefficients of the powers of $e_2$ (i.e. the terms in the representation in the base $\{e_1,e_2,e_3\}$ which have the form $c e_2^k$).