Going back to my Canadian Trucker comment, the reason the truck up the hill was in the ditch was lack of chains on the trailer, and the supra on the road, and the slow speed the truck was making going uphill.
In the American and Canadian west, when you need chains on the drivers you are required to apply them to the trailer.
If you tow in snow and the temps are cold, down below 10deg F, the snow will act a bit like pavement, especially if the road authority uses a serrated blade to grade the snow-covered road. The serrations help hold the road for your tires.
If temps rise toward the freezing point, friction heating from traffic will create a surface film of free water and make the road very slippery.
The worst situation is when the snow falls and packs and then the weather warms enough to put an ice film on the surface and then gets quite cold. The snow and iis now a bit of a skating rink, and normal tire chains just slide across the top.
They make chains with "V" bits welded on to face down and protrude below the links for the specific purpose of providing a bite into the ice/snow surface.
So the issues you need to evaluate are 1) Is it cold packed snow, is it warmed to icy cold packed snow, or is it warm and slippery snow. Next, (2) can your tow vehicle maintain sufficient speed to handle supra without gravity dragging you to the low side. Supra is the banking put into turns to enable higher speeds. Pleas note that this is an issue both uphill and downhill. And (3) are your trailer tires of equal or greater "grippiness" on snow as your truck tires. And lastly, (4) do you have independent control over your trailer brakes sufficient to apply them firmly but avoid lockup. Ideally, they should come on harder than the truck brakes in order to get the trailer in line, but not so hard as to lock up. Also, be careful when using the truck engine to reduce or control speed on the downhill, You brake system may work well with this, or not. Try to adjust it so it does. You want the force at the hitch to always be in tension, never in compression.