(* Matrix of scalings *)
W = {{0,w12,w13,w14,w15},{w12,0,w23,w24,w25},{w13,w23,0,w34,w35},{w14,w24,w34,0,w45},{w15,w25,w35,w45,0}};
(* Determinant of W *)
det = Det[W];
(* 4-minors of W *)
M = Minors[W,4];

(* ML degree 11 scaling *)
ReplaceAll[{w12->1,w13->1,w14->1,w15->1,w23->1,w24->1,w25->1,w34->1,w35->1,w45->1}][{det,M[[1]][[1]],M[[2]][[2]] ,M[[3]][[3]] ,M[[4]][[4]] ,M[[5]][[5]] }]

(* ML degree 10 scaling *)
FindInstance[det!=0&&M[[1]][[1]]==0&&M[[2]][[2]]!=0&&M[[3]][[3]]!=0&&M[[4]][[4]]!=0&&M[[5]][[5]]!=0&&w12≠0&&w13≠0&&w14≠0&&w15≠0&&w23≠0&&w24≠0&&w25≠0&&w34≠0&&w35≠0&&w45≠0,{w12,w13,w14,w15,w23,w24,w25,w34,w35,w45},Reals]
ReplaceAll[{w12->(7/8),w13->1,w14->1,w15->1,w23->1,w24->1,w25->1,w34->(32/7),w35->1,w45->1}][{det,M[[1]][[1]],M[[2]][[2]] ,M[[3]][[3]] ,M[[4]][[4]] ,M[[5]][[5]] }]

(* ML degree 9 scaling *)
FindInstance[det!=0&&M[[1]][[1]]==0&&M[[2]][[2]]==0&&M[[3]][[3]]!=0&&M[[4]][[4]]!=0&&M[[5]][[5]]!=0&&w12≠0&&w13≠0&&w14≠0&&w15≠0&&w23≠0&&w24≠0&&w25≠0&&w34≠0&&w35≠0&&w45≠0,{w12,w13,w14,w15,w23,w24,w25,w34,w35,w45},Reals]
ReplaceAll[{w12->1,w13->1,w14->1,w15->1,w23->1,w24->1,w25->1,w34->4,w35->4,w45->1}][{det,M[[1]][[1]],M[[2]][[2]] ,M[[3]][[3]] ,M[[4]][[4]] ,M[[5]][[5]] }]

(* ML degree 8 scaling *)
FindInstance[det!=0&&M[[1]][[1]]==0&&M[[2]][[2]]==0&&M[[3]][[3]]==0&&M[[4]][[4]]!=0&&M[[5]][[5]]!=0&&w12≠0&&w13≠0&&w14≠0&&w15≠0&&w23≠0&&w24≠0&&w25≠0&&w34≠0&&w35≠0&&w45≠0,{w12,w13,w14,w15,w23,w24,w25,w34,w35,w45},Reals]
ReplaceAll[{w12->1,w13->1,w14->1,w15->1,w23->1,w24->1,w25->1,w34->4,w35->4,w45->4}][{det,M[[1]][[1]],M[[2]][[2]] ,M[[3]][[3]] ,M[[4]][[4]] ,M[[5]][[5]] }]

(* ML degree 7 scaling *)
FindInstance[det!=0&&M[[1]][[1]]==0&&M[[2]][[2]]==0&&M[[3]][[3]]==0&&M[[4]][[4]]==0&&M[[5]][[5]]!=0&&w12≠0&&w13≠0&&w14≠0&&w15≠0&&w23≠0&&w24≠0&&w25≠0&&w34≠0&&w35≠0&&w45≠0,{w12,w13,w14,w15,w23,w24,w25,w34,w35,w45},Reals]
ReplaceAll[{w12->1,w13->1/4,w14->1,w15->1,w23->1,w24->1,w25->1,w34->1/4,w35->1/4,w45->4}][{det,M[[1]][[1]],M[[2]][[2]] ,M[[3]][[3]] ,M[[4]][[4]] ,M[[5]][[5]] }]

(* first ML degree 6 scaling *)
Simplify[FindInstance[det!=0 && M[[1]][[1]]==0&&M[[2]][[2]]==0&&M[[3]][[3]]==0&&M[[4]][[4]]==0&&M[[5]][[5]]==0&&w12≠0&&w13≠0&&w14≠0&&w15≠0&&w23≠0&&w24≠0&&w25≠0&&w34≠0&&w35≠0&&w45≠0,{w12,w13,w14,w15,w23,w24,w25,w34,w35,w45},Complexes]]
Simplify[ReplaceAll[{w12->-3,w13->(7/2),w14->(21/4) (1-I (Sqrt[3])),w15->(3/4) (1+I (Sqrt[3])),w23->-2,w24->-3-3 I (Sqrt[3]),w25->(3/7) I (I+(Sqrt[3])),w34->-7,w35->-1,w45->3}][{det,M[[1]][[1]],M[[2]][[2]] ,M[[3]][[3]] ,M[[4]][[4]] ,M[[5]][[5]] }]]

(* second ML degree 6 scaling *)
Simplify[FindInstance[det==0&&M[[1]][[1]]==0&&M[[2]][[2]]==0&&M[[3]][[3]]==0&&M[[4]][[4]]==0&&M[[5]][[5]]==0&&w12≠0&&w13≠0&&w14≠0&&w15≠0&&w23≠0&&w24≠0&&w25≠0&&w34≠0&&w35≠0&&w45≠0,{w12,w13,w14,w15,w23,w24,w25,w34,w35,w45},Complexes]]
Simplify[ReplaceAll[{w12->-4,w13->1,w14->4,w15->-(19/4)-2 (Sqrt[3]),w23->-1,w24->-4,w25->(19/4)-2 (Sqrt[3]),w34->4,w35->-4,w45->-3}][{det,M[[1]][[1]],M[[2]][[2]] ,M[[3]][[3]] ,M[[4]][[4]] ,M[[5]][[5]] }]]