The entanglement entropy of spacetime regions A in odd-dimensional conformal field theories (CFTs) contains a universal constant term, (-1)^((d-1)/2)F(A). This quantity can be robustly defined by considering the mutual information of pairs of slightly deformed versions of A. In the case of general three-dimensional CFTs, F(A) is positive definite and bounded below by the round disk result, F(A)≥ F₀≡ F(∂ A=S¹). Additionally, strong evidence has been provided that for every region A, F(A)/F₀ is maximized, within the space of CFT₃'s, by the free scalar field result. In this paper we show that while F(A) remains a local minimum around F₀≡ F(∂ A=S³) for small deformations of the spherical entangling surface, it can take values of arbitrarily large magnitude with either sign for more general regions, and hence it is neither upper- nor lower-bounded in general CFT₅'s. We argue that an analogous conjecture regarding the extremization of F(A)/F₀ for general regions within the space of theories fails in d=5. We instead analyze the viability of the weaker bound, F_ε/F₀≤ [F_ε/F₀]_(free scalar), ∀CFT₅ for general small geometric deformations of the spherical entangling surface. This is equivalent to a general constraint involving the stress-tensor two-point function C_T and the Euclidean partition function on the sphere, namely, C_T/F₀≤ [C_T/F₀]_( free scalar)≈ 0.314, which we show to hold for all known CFT₅'s. We also comment on possible extensions of this result to higher dimensions.