  The modified algebraic Bethe ansatz, introduced by Cramp\'e and the author
[8], is used to characterize the spectral problem of the Heisenberg XXZ
spin-$\frac{1}{2}$ chain on the segment with lower and upper triangular
boundaries. The eigenvalues and the eigenvectors are conjectured. They are
characterized by a set of Bethe roots with cardinality equal to $N$ the length
of the chain and which satisfies a set of Bethe equations with an additional
term. The conjecture follows from exact results for small chains. We also
present a factorized formula for the Bethe vectors of the Heisenberg XXZ
spin-$\frac{1}{2}$ chain on the segment with two upper triangular boundaries.
