Summary: We study the non-integral generalized Stieltjes constants \(\gamma_\alpha(a)\) arising from the Laurent series expansions of fractional derivatives of the Hurwitz zeta functions \(\zeta^{(\alpha)}(s, a)\), and we prove that if \(h_a(s) := \zeta(s, a) - 1/(s - 1) - 1/ a^s\) and \(C_\alpha(a) := \gamma_\alpha(a) - \frac{\log^\alpha(a)}{a}\), then \[ C_\alpha(a) = (- 1)^{- \alpha} h_a^{(\alpha)}(1), \] for all real \(\alpha \geq 0\), where \(h^{(\alpha)}(x)\) denotes the \(\alpha\)-th Grünwald-Letnikov fractional derivative of the function \(h\) at \(x\). This result confirms the conjecture of R. Kreminski [Math. Comput. 72, 1379–1397 (2003; Zbl 1033.11043)], originally stated in terms of the Weyl fractional derivatives.