Linear programming with parameters

Question

I am trying to run LinearProgramming with a vector and a matrix that depend on some parameters, but it's not working. The message I get is that my code "contains elements that are empty matrices, invalid vectors or matrices, or not real numbers". I suppose the problem is that Mathematica doesn't know that my entries are real because of the parameters. I'd rather not specify values. Here is my code:

α := -{0, -1152  c k^3 π^2, 
576  c k^3 π, -13824 k^4 π^3, 
576  k^4 π^2 (12 + c π), -24  k^4 π (288 + 
   c π (48 + c π)), -864 c k^3 - 72 c^2 k^3 π - 
 2 c^3 k^3 π^2, 
k^4 (10368 + c π (36 + c π) (72 + c π)) };

G := {{0, 0, 0, -8 k π^4 + 64 π^5 q^2, 
4 k π^3 - 32 π^4 q^2, -4 k π^2 + 48 π^3 q^2, 0, 
6 k π - 96 π^2 q^2}, {0, 8  π^2, -4  π, 
0, -4  k π^3, 8 k π^2 - 48 π^3 q^2, 
6, -18 k π + 144 π^2 q^2}, {0, 0, 0, 0, 
0, -2 k π^2 + 16 π^3 q^2, -6, 
9 k π - 48 π^2 q^2}, {0, 0, 0, 0, 0, 0, 2, -k π}};

$Assumptions = ({c, k, q} ∈ Reals)

LinearProgramming[α, G, {0, 0, 0, 0}, -Infinity]

I tried Refine, Assumptions, but none worked. What else can I do?