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In 2001, Zhongmin Shen asked if it is possible for two projectively relatedFinsler metrics to have the same Riemann curvature tensor, [14, page 184]. Inthis paper, we provide an answer to this question, within the class of Finslermetrics of scalar flag curvature. In Theorem 3.1, we show that the answer isnegative, for non-vanishing scalar flag curvature. The answer is known to bepositive when the scalar flag curvature vanishes, [12, 14] and this positiveanswer is related to the existence of many solutions to Hilbert's FourthProblem.  As a generalisation of this problem, we can ask if it is possible for a givenspray, with non-vanishing scalar flag curvature, to represent, afterreparametrisation, the geodesic spray of a Finsler metric. In Proposition 3.3,we show how to construct sprays whose projective class does not contain anyFinsler metrizable spray with the same Riemann curvature tensor.

Funk functions and projective deformations of sprays and Finsler spaces of scalar flag curvature