Let \({\mathcal H}_3\) be the three dimensional complex Heisenberg Lie algebra consisting of strictly upper triangular \(3\times 3\) matrices over \(\mathbb{C}\). Let \(e = z_{12}\), \(f= z_{23}\), \(x= z_{13}\) be the standard basis of \({\mathcal H}_3\), where \(z_{ij}\) is the matrix with 1 as the \((i,j)\)th entry and 0's elsewhere. For any \(k\in\mathbb{N}\), the truncated Lie algebra \(L_k:= {\mathcal H}_3\otimes \frac{\mathbb{C}[t]}{(t^{k+1})}\) acquires an \(\mathbb{N}^3\)-grading by assigning degrees (1,0,0), (0,1,0) and (0,0,1) respectively to \(e\), \(f\) and \(x\). This gives rise to an \(\mathbb{N}^3\)-grading on the Lie algebra homology \(H_*(L_k)\). In the paper under review, the authors prove that the Poincaré polynomial \[ \sum_{m,n\in\mathbb{N}}\dim H_{m,n,0}(L_k)u^m v^n = (1+u+v)^{k+1}. \] In particular, the 0-th \(x\)-graded piece of \(H_*(L_k)\) has dimension \(3^{k+1}\). The total homology of \(L_k\) is still unknown.