Suppose m and n are integers such that 1≤m≤n, and H is a subgroup of the symmetric group S of degree m. Define the generalized matrix function associated with the principal character of the group H on an m×m matrix B=(b) byd(B)=∑∏ b,and define the generalized numerical range of an n×n matrix A associated with d byW(A)=(d (V*AV):Visn×msuchthatV*V=I).It is known that W(A) is convex if m=1 or if m=n=2. Hu, Hurley and Tam made the following conjecture: Suppose H=S, 2≤m≤n with (m,n)≠(2,2). Let A∈M be a normal matrix. Then W(A) is convex if and only if A is a multiple of a Hermitian matrix. In this note, counter examples are given to show that the conjecture is not true when m