Transposition table based cutoff: Why not use betaOrig

Question

In this wiki entry following implementation is given

function negamax(node, depth, α, β, color) is
    alphaOrig := α

    (* Transposition Table Lookup; node is the lookup key for ttEntry *)
    ttEntry := transpositionTableLookup(node)
    if ttEntry is valid and ttEntry.depth ≥ depth then
        if ttEntry.flag = EXACT then
            return ttEntry.value
        else if ttEntry.flag = LOWERBOUND then
            α := max(α, ttEntry.value)
        else if ttEntry.flag = UPPERBOUND then
            β := min(β, ttEntry.value)

        if α ≥ β then
            return ttEntry.value

    if depth = 0 or node is a terminal node then
        return color × the heuristic value of node

    childNodes := generateMoves(node)
    childNodes := orderMoves(childNodes)
    value := −∞
    for each child in childNodes do
        value := max(value, −negamax(child, depth − 1, −β, −α, −color))
        α := max(α, value)
        if α ≥ β then
            break

    (* Transposition Table Store; node is the lookup key for ttEntry *)
    ttEntry.value := value
    if value ≤ alphaOrig then
        ttEntry.flag := UPPERBOUND
    else if value ≥ β then
        ttEntry.flag := LOWERBOUND
    else
        ttEntry.flag := EXACT
    ttEntry.depth := depth
    transpositionTableStore(node, ttEntry)

    return value

I think one could use betaOrig instead of β when storing TT entry. If β < betaOrig, β is an upper bound of the exact value of the node. If a child node fails low (child's value <= -β), fail-low nodes return an upper bound of the exact value, thus β <= -child's value <= -child's exact value <= exact value of current node <= β. It follows that β is the exact value of current node.

This means that it is harmless to loosen the inequality from value ≥ β to value ≥ betaOrig. Why doesn't it use betaOrig?