If, by reason of death, resignation, removal from office, inability, or failure to qualify, there is neither a President nor Vice President to discharge the powers and duties of the office of President, then the Speaker of the House of Representatives shall, upon his resignation as Speaker and as Representative in Congress, act as President.

...

If, at the time when under subsection (a) of this section a Speaker is to begin the discharge of the powers and duties of the office of President, there is no Speaker, or the Speaker fails to qualify as Acting President, then the President pro tempore of the Senate shall, upon his resignation as President pro tempore and as Senator, act as President.

https://www.law.cornell.edu/uscode/text/3/19

There's obviously no precedent for how this would work in practice. It kind of sounds like the Speaker could refuse to resign his position, thus "failing to qualify" and letting it fall to the next in line. It's unclear to me if his resignation is compelled by this language.