A few years ago this unassuming little collection found itself in possible possession of a blockbuster masterpiece, when it was suggested that its painting “Portrait of a Gentleman” was done by Caravaggio, the wildly popular art star of the late 16th- and early- 17th centuries. Many art historians remain skeptical, though a nifty digital display next to the painting makes a compelling case for the provenance. The museum’s other showpiece is a transcendently glowing portrait of Montepulciano’s patron saint, St. Agnes, by Domenico Beccafumi, who did much of the flooring in his Siena’s Duomo.