Equivalence of Lebesgue and Henstock-Kurzweil (Gauge) integral.

Question

1. If $f$ is Henstock-Kurzweil integrable $\Longrightarrow$ $f$ is measurable.
2. $f$ is Lebesgue integrable $\Longleftrightarrow$ $|f|$ is Henstock-Kurzweil integrable.
3. $|f|$ is Henstock-Kurzweil integrable $\Longrightarrow$ $f$ is Henstock-Kurzweil integrable.

Can we use the above three relations to show that $f$ is Lebesgue integrable $\Longrightarrow$ $f$ is Henstock-Kurzweil integrable, that is, can we drop the condition $f\ge 0$ as shown here ?

In a nutshell does there exits functions that are not Henstock-Kurzweil integrable but are Lebesgue integrable?