From the maps they show, it’s pretty clearly a great circle (the obvious meaning for “straight line” on the surface of a sphere), not constant bearing.

Whoops, turns out I didn't actually understand what "constant bearing" meant. What I actually meant was "no rudder/steering" which is exactly what people would imagine as travelling in a straight line.

A constant bearing actually gives a Rhumb line: https://en.wikipedia.org/wiki/Rhumb_line

Don't apologize. Your use is perfectly correct, you're obviously not referring to compass bearing.

Obvious perhaps, in a mathematically idealised system that's idealised according to axioms that are unstated ... a non-equitorial latitudinal line on a sphere is straight.

Walk (or drive, or sail) without turning and you’ll follow a great circle.

Stay perpendicular to North/South and you follow a parallel.

None of these things is of course possible IRL.

Staying perpendicular to north/south requires turning, unless you’re on the equator.

Or listing (ie leaning); still it seems straight, YMMV.

Consider the degenerate case where you’re ten feet away from the North Pole. You’ll need to go in a tight circle. The situation at more moderate latitudes is the same, just less extreme. It only seems straight because we’re usually at moderate latitudes and the rate of turning is low.

What sort of constant bearing - magnetic bearings will vary along a map-based bearing. There's an arbitrary choice involved. We're probably not looking at changes in sea-level that put us off line - is a necessarily idealised system in which the question makes sense.

But, unless you are passing over a pole, a great circle doesn't use a constant bearing.