I am interested in the dimension, or, even better, in the $S_8$-module structure of the space of degree 8 multilinear operations on Jordan algebras.

Recall that a Jordan algebra is a commutative but not generally associative algebra in which the product satisfies $$ (xx)(yx)=((xx)y)x, $$ in other words, multiplications by $x$ and by $x^2$ commute.

Elementary formulation: Let $J(x_1,\ldots,x_8)$ be the free Jordan algebra on eight generators, and let $V$ be the subspace of $J$ consisting of elements that have degree exactly one in each of the eight generators. What is the dimension of $V$, and how does $V$ decompose into irreducibles under the obvious $S_8$-action?

Operadic formulation: consider the operad $\mathcal{J}$ generated by one symmetric binary operation subject to the $S_4$-module of relation generated by the multilinearization of the Jordan identity. Concretely, that identity is $$ ((a_1a_2)a_3)a_4+((a_1a_4)a_3)a_2+((a_2a_4)a_3)a_1=(a_1a_2)(a_3a_4)+(a_1a_3)(a_2a_4)+(a_1a_4)(a_2a_3) . $$ Compute $\dim\mathcal{J}(8)$ and decompose that space as $S_8$-module.

Why 8? This question obviously depends on a positive integer parameter $n$. Up to $n=7$ the situation is well understood; the corresponding dimensions were computed by Glennie in late 1960s, are equal, respectively, to 1, 1, 3, 11, 55, 330, 2345, and are documented in the paper

C. M. Glennie, Identities in Jordan algebras, Computational Problems in Abstract Algebra. Pergamon, Oxford, 1970, pp. 307-313.

A scanned copy of that paper is stored on the OEIS website, where those dimensions are collected in the sequence A001776. The symmetric group decompositions in that range are also known, though it is harder to give a precise reference.

Of course, the case $n=8$ is the first "interesting" one, since this is where the story of "special" Jordan algebras begin. (A Jordan algebra is called special if it is a subalgebra of an associative algebra equipped with the operation $a\circ b=ab+ba$; there is, in addition to Jordan identities, an identity of degree $8$, also found by Glennie, that is satisfied only in special Jordan algebras.) However, one would expect that in the 50+ years that elapsed since Glennie's work, at least the step from $7$ to $8$ could have been made, and I wonder if anyone knows if that is indeed the case.

What I tried? I used two computational approaches, the Albert nonassociative algebra system and the operadic Gröbner basis calculator based on my own work, and alas my computer cannot go beyond $n=7$ in less than a day in either of them (and I did not leave it on for longer so far).

Many thanks for any possible leads!