All shortest paths from start vertex s to end vertex t for graphs with positive edge weights

Question

The code for finding all shortest paths from start vertex s to end vertex t is provided in BreadthFirstScan. The relevant parts are under Applications, then Shortest Path Applications, and in that section: the second example (for unweighted graphs).

Is there online info where an adaptation is given for graphs with edge weights (positive weights)?

The code for unweighted case is copied below to find all shortest paths in an unweighted graph, using the example of a grid graph:

s = 1; t = 3*4; 
g = GridGraph[{3, 4}, VertexSize -> {s -> Medium, t -> Medium}]

discoverFun[u_, v_, d_] :=  If[u != v,  PropertyValue[{g, u},      
"ShortestPaths"] =  Table[Append[p, u], {p, PropertyValue[{g, v}, 
"ShortestPaths"]}];  PropertyValue[{g, u}, "Distance"] = d]

revisitFun[u_, v_] :=  If[PropertyValue[{g, u}, "Distance"] ==     
PropertyValue[{g, v}, "Distance"] + 1,  PropertyValue[{g, u}, 
"ShortestPaths"] =  Join[PropertyValue[{g, u}, "ShortestPaths"],     
Table[Append[p, u], {p, PropertyValue[{g, v}, "ShortestPaths"]}]]]

PropertyValue[{g, s}, "ShortestPaths"] = {{s}}; 
PropertyValue[{g, s}, "Distance"] = 0;

BreadthFirstScan[g,  s, {"DiscoverVertex" -> discoverFun, 
"VisitedVertex" -> revisitFun,  "UnvisitedVertex" -> revisitFun}];

Table[HighlightGraph[g, p], {p,  PathGraph /@ PropertyValue[{g, t}, 
"ShortestPaths"]}]