Rinse cycle issue

I have a Whirlpool Washing Machine (LXR7133JQ1 model) that was bought in 2003. The hot/warm water works but anytime we use just cold water only it barely comes out with little water pressure. I turned the cold water valve on all the way from the house shut off valve to the washer and it still barely comes out in the washing machine. I took out and cleaned the screen in the cold water connection and it still has the same issue. Also I dont see any leaks. Please advise what would cause the cold water to not come out all the way in my wash machine.

If I turn the cold water house valve off and then on, the wash cold water rushes out and does work for several seconds then slows down with less pressure. Cold Water pressure is working everywhere in the house except for this washing machine.

Please advise.

 

Take the cold water hose off from back of washer. Use a small screwdriver and remove screen from valve in washer. if clogged clean it, if not you will need a water mixer valve. While hose is off put end in basket and check to see if there is water pressure from hose.

 

I have this intuition that Kensington already found likely cause due to all the other stuff he tried.

Invariably if the pressure is good, then slows, there is a restriction upstream. If at the fill-valve or downstream, the trickle should exist from the beginning. Water pressure is high at first because the hose, which will expand under pressure, and fill-valve, are likely under full pressure and volume capability until say a restriction (in the shut-off valve, from say a displaced washer or chunk, or nearly rusted- closed galvanized piping in that area of the house) causes the volume to slow.