Hu and Ye conjectured that for a $k$-th order and $n$-dimensional tensor $\mathcal{A}$ with an eigenvalue $λ$ and the corresponding eigenvariety $\mathcal{V}_λ(\mathcal{A})$, $$\mathrm{am}(λ) \ge \sum_{i=1}^κ\mathrm{dim}(V_i)(k-1)^{\mathrm{dim}(V_i)-1},$$ where $\mathrm{am}(λ)$ is the algebraic multiplicity of $λ$, and $V_1,\ldots,V_κ$ are all irreducible components of $\mathcal{V}_λ(\mathcal{A})$. In this paper, we prove that if $\mathcal{A}$ is a nonnegative weakly irreducible tensor with spectral radius $ρ$, then $\mathrm{am}(λ) \ge |\mathbb{V}_λ(\mathcal{A})|$ for all eigenvalues $λ$ of $\mathcal{A}$ with modulus $ρ$, where $\mathbb{V}_λ(\mathcal{A})$ is the projective eigenvariety of $\mathcal{A}$ associated with $λ$. Consequently we confirm Hu-Ye's conjecture for the above eigenvalues $λ$ of $\mathcal{A}$ and also the least H-eigenvalue of a weakly irreducible $Z$-tensor. We prove several equality cases in Hu-Ye's conjecture for the eigenvalues of the adjacency tensor or Laplacian tensor of uniform hypergraphs.