How to handle 'line-continuation' using parser combinators

I'm trying to write a small parser using the Sprache parser combinator library. The parser should be able to parse lines ended with a single \ as insignificant white space.

Question

How can I create a parser that can parse the values after the = sign that may contain a line-continuation character \? For example

a = b\e,\
    c,\
    d

Should be parsed as (KeyValuePair (Key, 'a'), (Value, 'b\e, c, d')).

I'm new to using this library and parser combinators in general. So any pointers in the right direction are much appreciated.

What I have tried

Test

public class ConfigurationFileGrammerTest
{
    [Theory]
    [InlineData("x\\\n  y", @"x y")]
    public void ValueIsAnyStringMayContinuedAccrossLinesWithLineContinuation(
        string input, 
        string expectedKey)
    {
        var key = ConfigurationFileGrammer.Value.Parse(input);
        Assert.Equal(expectedKey, key);
    }
}

Production

    public static readonly Parser<string> Value =
        from leading in Parse.WhiteSpace.Many()
        from rest in Parse.AnyChar.Except(Parse.Char('\\')).Many()
            .Or(Parse.String("\\\n")
            .Then(chs => Parse.Return(chs))).Or(Parse.AnyChar.Except(Parse.LineEnd).Many())
        select new string(rest.ToArray()).TrimEnd();

Xunit.Sdk.EqualException: Assert.Equal() Failure
           ↓ (pos 1)
Expected: x y
Actual:   x\
           ↑ (pos 1)

    public static readonly Parser<string> SingleLineValue =
        from leading in Parse.WhiteSpace.Many()
        from rest in Parse.AnyChar.Many().Where(chs => chs.Count() < 2 || !(string.Join(string.Empty, chs.Reverse().Take(2)).Equals("\\\n")))
        select new string(rest.ToArray()).TrimEnd();

    public static readonly Parser<string> ContinuedValueLines =
        from firsts in ContinuedValueLine.AtLeastOnce()
        from last in SingleLineValue
        select string.Join(" ", firsts) + " " + last;

    public static readonly Parser<string> Value = SingleLineValue.Once().XOr(ContinuedValueLines.Once()).Select(s => string.Join(" ", s));

Xunit.Sdk.EqualException: Assert.Equal() Failure
           ↓ (pos 1)
Expected: x y
Actual:   x\\n  y
           ↑ (pos 1)