  Critical site percolation on the triangular lattice is described by the
Yang-Baxter solvable dilute $A_2^{(2)}$ loop model with crossing parameter
specialized to $\lambda=\frac\pi3$, corresponding to the contractible loop
fugacity $\beta=-2\cos4\lambda=1$. We study the functional relations satisfied
by the commuting transfer matrices of this model and the associated Bethe
ansatz equations. The single and double row transfer matrices are respectively
endowed with strip and periodic boundary conditions, and are elements of the
ordinary and periodic dilute Temperley-Lieb algebras. The standard modules for
these algebras are labeled by the number of defects $d$ and, in the latter
case, also by the twist $e^{i\gamma}$. Nonlinear integral equation techniques
are used to analytically solve the Bethe ansatz functional equations in the
scaling limit for the central charge $c=0$ and conformal weights
$\Delta,\bar\Delta$. For the groundstates, we find $\Delta=\Delta_{1,d+1}$ for
strip boundary conditions and
$(\Delta,\bar\Delta)=(\Delta_{\gamma/\pi,d/2},\Delta_{\gamma/\pi,-d/2})$ for
periodic boundary conditions, where $\Delta_{r,s}=\frac1{24}((3r-2s)^2-1)$. We
give explicit conjectures for the scaling limit of the trace of the transfer
matrix in each standard module. For $d\le8$, these conjectures are supported by
numerical solutions of the logarithmic form of the Bethe ansatz equations for
the leading $20$ or more conformal eigenenergies. With these conjectures, we
apply the Markov traces to obtain the conformal partition functions on the
cylinder and torus. These precisely coincide with our previous results for
critical bond percolation on the square lattice described by the dense
$A_1^{(1)}$ loop model with $\lambda=\frac\pi3$. The concurrence of all this
conformal data provides compelling evidence supporting a strong form of
universality between these two stochastic models as logarithmic CFTs.
