How to find the exact optimal bound

Question

Is there any way to fix or revise this code to find the EXACT optimal bound, and could anyone help find the EXACT optimal bound? Thanks.

generateRandomUnitVector[] := Normalize[RandomReal[{-1, 1}, 5]];

deleteRowsCols[mat_, {i_, j_}] := Module[{dim},
  dim = Length[mat];
  Part[mat, Complement[Range[dim], {i, j}], Complement[Range[dim], {i, j}]]
];

findMinimalUpperBound[numSamples_] := Module[{minEigenValuesList = {}, x, y, z, M, subMatrices, minEigenValues},
  For[n = 1, n <= numSamples, n++,
    x = generateRandomUnitVector[];
    y = generateRandomUnitVector[];
    z = generateRandomUnitVector[];
    
    M = x.x\[Transpose] + y.y\[Transpose] + z.z\[Transpose];
    
    subMatrices = Table[deleteRowsCols[M, {i, j}], {i, 1, 5}, {j, i + 1, 5}];
    
    minEigenValues = Min[Table[Min[Eigenvalues[subMatrices[[i, j - i]]]], {i, 1, 4}, {j, i + 1, 5}]];
    AppendTo[minEigenValuesList, minEigenValues];
  ];
  
  Max[minEigenValuesList]
];

minimalUpperBound = findMinimalUpperBound[1000];
minimalUpperBound

(The above code defines a function to generate random unit vectors in R^5 and Function to delete ith and jth rows and columns from a matrix, ...)