  We study a homogeneous system of $d+8$ linear partial differential equations
(PDEs) in $d$ variables arising from two-dimensional Conformal Field Theories
(CFTs) with a $W_3$-symmetry algebra. In the CFT context, $d$ PDEs are third
order and correspond to the null-state equations, whereas the remaining 8 PDEs
(five being second order and three being first order) correspond to the $W_3$
global Ward identities. In the case of central charge $c=2$, we construct a
subspace of the space of all solutions which grow no faster than a power law.
We call this subspace the space of $W_3$ conformal blocks, and we provide a
basis expressed in terms of Specht polynomials associated with column-strict,
rectangular Young tableaux with three columns. The dimension of this space is a
Kotska number and it coincides with CFT predictions, hence we conjecture that
it exhausts the space of all solutions having a power law bound. Moreover, we
prove that the space of $W_3$ conformal blocks is an irreducible representation
of a certain diagram algebra defined from $\mathfrak{sl}_3$ webs that we call
Kuperberg algebra. Finally, we formulate a precise conjecture relating the
$W_3$ conformal blocks at $c=2$ to scaling limits of probabilities in the
triple dimer model recently studied by Kenyon and Shi. We verify the conjecture
for explicit examples up to $d=6$. For more general central charges, we expect
that $W_3$ conformal blocks are related to scaling limits of probabilities in
lattice models based on $\mathfrak{sl}_3$ webs.
