I wonder if you’ve got a weak fuel pump that can’t keep up with the increasing demand under load.
If you have a gap tool, you could monitor your fuel rail pressure and demanded rail pressure when driving and then climbing a hill. Actual pressure and demanded pressure should stay pretty well aligned. If demanded pressure is signficantly higher, that suggests a fuel supply issue.
A weak pump also should show up in fuel trims, certainly with load but maybe even without. With a weak fuel pump, fuel trims should show increasingly positive trim as rpm or load increases. If you check fuel trim, make sure to look at both short and long term trims and add the two together to get overall trim data. Might start with checking trims while stationary (in neutral or park) at idle, 1500 rpms, and 2500 rpms to see what it looks like. Monitoring fuel trims while driving is a bit tricky because the short term trim is always bouncing around.
Note that these tests won’t tell you whether the problem is the high pressure or low pressure pumps, but might point you in the direction of a fuel supply issue.