Coming with a late update: I bought another module like the initial one, installed a 47uF electrolytic cap at the input and am trying to power it with 17V.
With no load, the output stays at around 15V (14.85V, to be exact).
However, as soon as I apply a load of 50 mA, the output goes up to 16.36V.
If I then increase the input voltage, the output tracks the input, until I get to about 18.9V on the input, when the output voltage (under the 50 mA load) becomes exactly 15.00V.
From 18.9V up to 20V on the input (I didn't have the courage to go above that), the output voltage stays the same (i.e. doesn't track the input voltage anymore).
Initially, I thought maybe this was due to the module coming off eBay and the possibility the LT3042 in it might not be genuine.
But (and this is why I took so long for the update), I finally managed to order an LT3042 IC off DigiKey and install it on the board of the initial module from eBay (the one which prompted this discussion) and the behaviour is the same as with my 2nd module from eBay.
I am bit confused at this stage and not sure what I'm doing wrong here.