Consecutive numbers of the form $n^m + m^{n + 1}$

Question

Playing around with some numbers, I recently noticed that $2024 = 2^{10} + 10^3$ and $2025 = 3^6 + 6^4$. Are there more examples of positive integers $k$ such that both $k$ and $k + 1$ can be expressed as $n^m + m^{n + 1}$ for positive integers $n, m \geq 2$? I searched many small cases but could not find any, leading me to wonder if one could prove that $2024$ is the only such number. There doesn't seem to be any clear way for me to approach this.