Laissez-vibrer tie of non-zero duration?

[Kalirren]

I'm attempting to typeset a piece of music that contains a laissez-vibrer tie
that starts from a note, stretches over rests, and ends on a beat on which
there is a rest in the same voice as the initial note.  This last
requirement seems to rule out the use of a hidden note.

The output I am looking for is similar to that which be gotten from 

\relative c' {
\slurDown e8 ( r16 r8.)
}

except that the tie is completely horizontal.

Does anyone know how to manipulate the \laissezVibrer function to make this
happen, or of some other workaround?  In general, is it possible to create a
laissez-vibrer tie that lasts a certain length of time, or until a certain
note?
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