Does each integer have the form $x^4-y^3+z^2$ with $x,y,z$ positive integers?

Question

Let $\mathbb Z^+$ denote the set of positive integers. Here I ask the following question.

QUESTION: Does each integer have the form $x^4-y^3+z^2$ with $x,y,z\in\mathbb Z^+$?

I guess that the answer is affirmative. I have verified that any integer $m$ with $|m|\leqslant 10^5$ can be written as $x^4-y^3+z^2$ with $x,y,z\in\mathbb Z^+$ (cf. http://oeis.org/A266152). For example, \begin{gather*}0=4^4-8^3+16^2,\ \ -1=1^4-3^3+5^2,\\-20=32^4-238^3+3526^2,\ \ 11019=4325^4-71383^3+3719409^2.\end{gather*}

Moreover, based on some heuristic arguments, I conjecture that if $\{a,b,c\}$ is among the multisets $$\{2,3,3\},\ \{2,3,4\},\ \{2,3,5\},$$ then any integer $m$ can be written as $x^a+y^b-z^c$ with $x,y,z\in\mathbb Z^+$ in infinitely many ways. See Conjecture 5.1 and Remark 5.1 of my paper New conjectures on representations of integers (I).

I don't know how to prove that the question has a positive answer. Any comments on the question are welcome!