Products of diagonalizable triangular matrices

It is known that a $n\times n$ matrix defined over a field can be expressed as a product of diagonalizable matrices. In this work, this problem is studied for triangular matrices of sizes $n\times n$ ($n\in\mathbb{N}$) and $\mathbb{N}\times\mathbb{N}$. It is proved that each such matrix that is invertible can be expressed as a product of at most four diagonalizable triangular matrices. Additionally, some partial results for noninvertible matrices are presented.