Lighting, Light Fixtures, Ceiling and Exhaust Fans - Trouble wiring three-way switch

I am a first-timer finishing my basement, and so far I have done very well. I have hit one snag, however. I have been unable to correctly wire a three-way switch at the stairway to my basement. This is the setup:

Single gang box (Switch 1-- SW1)
Double gang box (Switch 2-- SW2; Switch 3-- SW3)
Stairway light

SW1 and SW2 are three-way. SW3 is a single pole. SW3 controls the light to the utility room (on the left at the bottom of the stairs). SW1 and SW2 are supposed to control the light at the bottom of the stairs. Coming into the double gang box are:

1. 14/3 wire between SW1 and SW2
2. 14/2 wire to stairway light
3. 14/3 wire to SW3 (although it's a single pole, it controls a light and an always-hot receptacle, requiring a 14/3 wire to the light)
4. 14/2 wire to power source

The wiring I have right now in the double gang box is:

I have spliced all the black wires together with two pigtails. One pigtail is connected to the common terminal of SW2 and one is connected to a hot terminal of SW3. All the white wires are spliced together, except for the one from (1), which is connected to the traveller terminal of SW2. All the grounds are spliced and pigtailed to each switch. The red wire from (1) is connected to the other traveller terminal of SW2. The red wire from (3) is connected to the other hot terminal of SW3.

All this does is makes the stairway light always-hot, while SW1 and SW2 do nothing. I am looking for a solution which I can implement without changing the wiring to the light, since the stairway was finished when we bought the house and I don't feel at all like tearing up the drywall and cramming myself into the tiny space there.

Any suggestions? Thanks in advance.

Of course the stairway light is always hot. You connected it's hot wire to all the other hot wires.

The black wire for the stairway light needs to be connected to the black wire coming back from SW1. Those two wires get connected together, but not to anything else.

When you have any wiring situation involving switches, especially multiple ones, try drawing pictures. If you draw a picture you can easily see why your solution doesn't work. When you do draw pictures, omit the ground wires to make the pictures easier to follow.

Thank you! That solved the problem. It seems so obvious now, I'm not sure why I didn't see it sooner. Sometimes it just takes a fresh set of eyes; I think I was over-analyzing and thinking too much about the problem and missed the forest for the trees.

Thanks again!