  The specific relative entropy, introduced by N. Gantert, allows to quantify
the discrepancy between the laws of potentially mutually singular measures. It
appears naturally as the large deviations rate function in a randomized version
of Donsker's invariance principle, as well as in a novel transport-information
inequality recently derived by H. Foellmer. A conjecture, put forward by the
aforementioned authors, concerns a closed form expression for the specific
relative entropy between continuous martingale laws in terms of their quadratic
variations. We provide a first partial result in this direction, by
establishing this conjecture in the case of well-behaved martingale diffusions
on the line.
