Depending where the washer machine drain is in respect to the entire 'system', you can try to snake from another point. If this drain is UPSTREAM from, say, a toilet, you can drain and remove the toilet to see if you get different results. But if this drain is, basically, running right into the sewer, you can try a longer snake.
A test you can do is plug and fill the utility sink. (Good thing you have one.) Mark the height you start at. Then, pull the plug. Note the water level where the water starts to slow up or stop. That will give you and the plumber an idea of how far the blockage might be. YOu have to realize that once the water leaves the utility sink's drain pipe, it's probably connected to a 4" lateral. (Really, anywhere from a 3" to an 8" lateral, depending.) In short, you have water from a 'narrow' pipe running into a pipe that's bigger so the pressure at the 'blockage' is nominal. Kinda like the hour glass phenomena where the top is full of sand but it dribble through a very small opening into a cavity whose volume is LARGER than the opening. Hence, nominal back-pressure. Nonetheless, you can still get an idea of how far down the line the blockage is.