Property of the symmetric reduction of a multivariate polynomial

Question

The following symmetric multivariate polynomial:

$$P(x_1,\ldots,x_n)=\prod_{1 \le i \lt j \le n}{\left(1+x_{i}x_{j}\right)}$$

has the following property: the number of classes of its monomials, defined as the sets of monomials that can be made equal to one another with a permutation of variables, is OEIS A004251. This is justified here (see the bottom of that question).

For $n=2,3,4,5$ I noticed that the number of monomials of the symmetric reduction of $P(x_1,\ldots,x_n)$ expressed with each elementary symmetric polynomial as a variable, results in the same above OEIS sequence. See the $n=5$ case, the the $n=4$ case and the the $n=3$ case.

Is this true for all values of $n$? And how could it be proved?

I noticed that in general symmetric polynomials don't have this property, for example the symmetric reduction of $x^3+y^3+z^3$ has $3$ monomials, not $1$.